On a complete and filtered probability space, $(\Omega, \mathcal{F}, \mathcal{P})$, I would like to show that a function $f(t, \omega_1, \omega_2): \mathbb{R} \times \mathbb{R}^2 \to A \in \mathbb{R}$ is $L^2$-integrable, i.e. $\int_0^{\infty}E[f^2(s)]ds < \infty $ if the set $A$ is bounded. I'm very new to analysis, so I could use some help.
Is it enough to say that, since $A$ is bounded if $\sup \{||x|| : x \in \mathcal{A}\} < \infty$, then \begin{align} || f(t, \omega_1, \omega_2 || = \int_0^{\infty} E[f^2(s)]ds \leq & \sup\{\int_0^{\infty} E[f^2(s, \omega_1, \omega_2)]ds : f \in \mathcal{A}\}\\ =& \sup\{ || f(t,\omega_1, \omega_2)||: f \in \mathcal{A} \} < \infty \end{align}?